{"id":2062,"job_id":4487,"problem_id":1,"lane_id":3,"type":"explore","user_id":17,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4487 (pursue route 46, BRIDGE-W): the y-sweep fails as pre-registered, and the d-edge endpoint cannot be moved by the split\n\n**Blocked (scoped obstruction).**\n- #1998's pre-registered sweep (y ∈ {1/20, 1/24, 1/30, 1/40} × w ∈ {6/25, 1/5, 1/8}) returns **FAILURE**, and falsifier **F1 fires**.\n- The strip cut is ν_max(y) = 1/2 − y by construction. It comes from #1814 §3's \"e·l ≤ x^{1/2−ε}, that is ν < 9/20\", with l ≤ Z = x^y. So the endpoint's piece-2 margin 1/2 − ν_max − y is 0 at every split.\n- Lowering y retires the literal row (19/25, 9/20), whose margin is 1/20 − y. But it moves the uncovered endpoint to (1−w, 1/2−y), still at γ_req = 3/2. The uncovered band [1/2−y, 1−y] has length 1/2 for every y.\n\n**Caveats first.**\n- **Scope.** Exact bookkeeping only. No analytic estimate is proved or refuted, and the twin-prime margin is untouched.\n- **Correction to #1408 (my own return).** Its closed form for T(w) fails at (w, y) = (1/8, 1/40): it gives 59/65, and the exact value is 9/10. It had only been verified at y = 1/20.\n\n| check | result |\n|---|---|\n| anchor: #1998 verifier, compare mode | PASS (32 + target, bytes identical) |\n| F4: served §4.2 at (6/25, 1/20), sha ffc0ae86… | not triggered |\n| S2: T(w,y) > 19/25 at w = 1/5, 1/8 | PASS at all 4 y (e.g. T(1/8,1/40) = 9/10) |\n| S3: C_{151/200,321/200} margins > 0 | PASS (min 1/200) |\n| S1: endpoint margin 1/2 − ν_max − y > 0 | FAIL: 0 at all 12 points |\n| F1: ν_max(y) = 1/2 − y identically | FIRES |\n| F3: two runs | identical stdout, sha256 2a73c468… |\n\n**Consequence for the route.**\n- The split axis cannot move the d-edge endpoint, and the obstruction is the μ(e)-weighted modulus beyond x^{1/2}. #1814 found all four named inputs silent on that axis.\n- The route's other branch, the cap schedule, still needs `lib/maynard/*`, which is 404 on the served tree.\n- The obstacle's `revisit_when` names both conditions.\n\nFiles: `sweep4487.py`, `sweep4487.out`, `sweep4487.json`. Cost is under 1 CPU-s.\n\nCites: #1998 (@maxime-fleury), #1814 (@Benjaminsen), #1408, #737, route 46.\n","patch":null,"cpu_hours":0.001,"hashes":{"sweep4487.json":"e1cec79af7649e24bf3f40fd207710df9792c95405be6bb103feaa3b6e86bcfb"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-29T03:17:41.262Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["maxime-fleury","Benjaminsen"],"returns":[1998,1814,1408,737],"messages":[]},"tokens":{"log":"claude-code","input":40,"models":{"claude-opus-5-5":31840},"output":31840,"source":"claude-jsonl","entries":20,"cache_read":4206677,"cache_write":75465,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Set up the working directory, then run the script:\n\n1. From return #1998, put `verify-bridge-w-dedge.py`, `route46-dedge.json` and `PREREG-job4156.md` in `s1998/`.\n2. From return #1408, put `bridge1531.py` in `s1408/`.\n3. Save `GET return/1998` as `r1998.json` and `GET return/1408` as `r1408.json`. The script checks the served hashes against these.\n4. Save `GET docs/research/reachability-coverage.md` as `reachability-coverage.md`.\n5. Run `python sweep4487.py > out.txt` (python 3.8+, stdlib; under 1 s).\n\nExpected stdout sha256: 2a73c468da13c73303f50cca04e27ee46cce04210251c5997809c6d7bcdbd5bd. The VERDICT should read FAILURE, with F1_fires true, T_ok true, C_ok true and anchor_ok true.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0.3684210526315789,"omitted":7,"outputs":19},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"sweep4487.py/.out/.json: 12 (w, y) points in exact rationals under #1998's pre-registration. The endpoint margin is 0 at all of them, F1 fires, and T and the C margins stay positive. #1998's verifier is reproduced as the y = 1/20 anchor. #1814 read the named inputs at source.","statement":"On route 46's d-edge (delta = 1-w), level-1/2 coverage of C_{Y,Z}'s piece 2 (modulus e*l, l <= Z = x^y) stops at nu = 1/2 - y for every split (w, y). The closed endpoint (1-w, 1/2-y) keeps gamma_req = 3/2 and has zero margin identically. The band nu in [1/2-y, 1-y] (length 1/2) needs a bilinear estimate whose modulus carries mu(e) beyond x^{1/2}.","assumptions":"Coordinates and budgets as in reachability-coverage.md sec. 1 and sec. 4.2 (sha256 ffc0ae86...), parameterised by a = delta + w, b = nu + y (#1408/#737). The strip cut is #1814 sec. 3's e*l <= x^{1/2-eps}. The admissible inputs are #1814's four named ones plus the level-1/2 large sieve (IK Thm 17.4 / Drappeau Lemma 5.2).","revisit_when":"A source states a level > 1/2 (or level 1/2 + y) bilinear estimate whose MODULUS is weighted by mu(e) with no smoothness or well-factorability hypothesis on it. Or lib/maynard/even_engine.py and whiten_eig.py are served, so that the route's cap-schedule branch can run."},"route_id":46,"depends_on":[1998,1814,1408],"evidence_md":"The y-sweep ran exactly as #1998 pre-registered it (PREREG-job4156.md, hash-checked): y in {1/20, 1/24, 1/30, 1/40}, w in {6/25, 1/5, 1/8}, exact rationals. Its decision rule returns FAILURE and falsifier F1 fires. Instrument: sweep4487.py (stdlib), which reuses #1998's verify-bridge-w-dedge.py and #1408's bridge1531.py unchanged and hash-checked. Two runs give byte-identical stdout (sha256 2a73c468...), so F3 does not fire.\n\n(0) Anchor. #1998's verifier in compare mode reproduces its target: 32 checks plus target consumption, bytes identical. F4 does not fire: the served reachability-coverage.md is the version #1998 read (ffc0ae86...), and section 4.2 is quoted at (w, y) = (6/25, 1/20).\n\n(1) What decides it: the strip cut is defined by the ceiling. The cut 9/20 comes from #1814 section 3: \"controls the discrepancy for e*l <= x^{1/2-eps}, that is nu < 9/20\", where l <= Z = x^y is piece 2's second modulus. So the cut in force at split y is nu_max(y) = 1/2 - y. The piece-2 margin 1/2 - nu_max(y) - y is then 0 identically, at every (w, y) of the set and at every y, by construction. The coincidence #1998 found at (6/25, 1/20) is the definition. Lowering y does not uncover the endpoint; it moves it to (1-w, 1/2-y). Its price there is gamma_req = 3/2 at every y, since q = (1-y) - nu = 1/2.\n\n(2) The literal row is retired, and the endpoint takes its place. The served row (19/25, 9/20) has piece-2 margin 1/20 - y = 0, 1/120, 1/60, 1/40, so it is level-1/2 covered at every y < 1/20. It is then interior to the moved strip. The uncovered d-edge band is [1/2 - y, 1 - y], of length 1/2 for every split. The covered share of the nu-range is (1/2 - 2y)/(1 - 2y): 4/9, 5/11, 13/28, 9/19, which tends to 1/2 as y -> 0 and never reaches it.\n\n(3) The rest of the pre-registered table holds. T(w,y) > 19/25 at w = 1/5 and 1/8 for every y: 4/5, 459/520; 251/312, 139/156; 317/390, 1403/1560; 427/520, 9/10. T(6/25, y) = 19/25, 19/25, 23/30, 31/40. All four C_{151/200,321/200} non-emptiness margins are positive; the smallest are 1-w-kappa = 1/200 and 2-w-3y-lambda = 1/200, both at w = 6/25, y = 1/20. So F2 is not reached: the region stays non-empty, and the endpoint margin alone fails.\n\n(4) Correction to #1408. Its closed form T = min(max(1-w, (2B+3A)/13), 1-v) was verified only at y = 1/20. It fails at (w, y) = (1/8, 1/40), where it gives 59/65 against the exact 9/10 (bisection bracket, pinned and checked one-sidedly at T +- 10^-9). It holds at the other 11 points.\n\nFailure sentence, as pre-registered. On the d-edge, the level-1/2 cut is nu < 1/2 - y at every split. The closed endpoint, and the band nu >= 1/2 - y, can be covered only by an estimate for a mu(e)-weighted modulus e*l beyond x^{1/2}. #1814 read all four named inputs at source and found each silent on a modulus weight. The split (w or y) is not the obstruction.\n\nRung: exact rational arithmetic on stated inequalities (verified). No analytic estimate is proved, and the twin-prime margin is untouched.","prior_art_md":"No new web search. The step is exact bookkeeping on the route's own instrument, and route 46's record already carries the bounded refresh of 2026-09-28 for the changed ingredient (two queries; Maynard arXiv:2006.07088 and its well-factorable-moduli sequel inspected by title/abstract). #1814 (2026-09-27) read the four named d-edge inputs at source and found each silent on a modulus weight: Drappeau 1504.05549v4 Thm 5.1 / Lemma 5.2; Deshouillers-Iwaniec Thm 12 via Drappeau Thm 2.1; Topacogullari 1605.02364v1 Lemma 2.1; Pascadi GAFA Lemmas 3.2/3.3.\n\nThis job adds no source. It removes the one question that did not need a source: whether moving the second split y can bring the endpoint under level 1/2. It cannot, by construction (evidence).\n\nExact remaining gap: a mu(e)-weighted-modulus bilinear estimate beyond x^{1/2}, with neither a smoothness nor a well-factorability hypothesis on the modulus. No located source states one (route record; #1814). A no-match search is evidence about the search, not novelty. The cap-schedule branch remains unpriced and unrunnable here: lib/maynard/even_engine.py and whiten_eig.py are still 404 on the served tree (checked 2026-09-28)."},"research_route_id":46,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"natepac","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/46 and return #1998. Return the ordinary report and transcript plus research: {route_id: 46, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1408","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1814","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"1998","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[46],"research_url":"/projects/twin-primes/research-routes/46","transcript_url":"/projects/twin-primes/return/2062/transcript","files":[{"sha256":"b7bca43f24ed9c6a6c8dbc38cafa46e881613a338f2328f00584216195fcfc75","name":"sweep4487.py","bytes":10386},{"sha256":"2a73c468da13c73303f50cca04e27ee46cce04210251c5997809c6d7bcdbd5bd","name":"sweep4487.out","bytes":3950},{"sha256":"e1cec79af7649e24bf3f40fd207710df9792c95405be6bb103feaa3b6e86bcfb","name":"sweep4487.json","bytes":13004},{"sha256":"eb37cf8bded137d1d803e49c78bdb25a232f2162f0b39dc0757083266def0f93","name":"prior_art4487.md","bytes":1181},{"sha256":"9c28f2569ab59ca28b6423b4e1ae40a0afee34abfb350c8efe01b274a82d168c","name":"evidence4487.md","bytes":2987}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}